The exact price of local realism in CHSH experiments: a measurement-dependence-detection trade-off surface, a moiré phase-locking mechanism that saturates it, and an unmeasured fringe in the fourfold coincidence sum

Abstract: For local hidden-variable accounts of the CHSH experiment that are faithful -- reproducing the observed singles and coincidence rates and unbiased marginals of a polarization singlet with symmetric detector efficiency $η<em>{\rm eff}$ -- I determine the minimal measurement dependence $M$ (Hall's variational measure) needed to achieve a CHSH value $S$. Linear programming over all local strategies yields, to machine precision at 32 grid points, $$M(S,η</em>{\rm eff})=\max{0,η<em>{\rm eff}((S+2)η</em>{\rm eff}-4)/6}$$, whose edges reproduce Hall's tight bound at $η<em>{\rm eff}=1$, the Garg-Mermin detection threshold, and the postselection ceiling $S=4/η</em>{\rm eff}-2$. The quantum point is certified exactly: $M(2\sqrt{2},9/10)=(27\sqrt{2}-33)/100$, with primal and dual certificates in $\mathbb{Q}(\sqrt{2})$ arithmetic. I solve the unique detection profile $D(m)=\sqrt{m}\,h(m)$ under which a deterministic sign model reproduces the singlet exactly, derive the $\sqrt{m}$ edge law, and prove exact quantum correlations and angle-independent coincidence rates jointly impossible for any pure-detection model of this class. Surviving local accounts trade off measurement dependence against a $\cos 4(a-b)$ modulation of the fourfold coincidence sum, of relative amplitude up to 12.4%, which vanishes identically at the CHSH angles and appears never to have been bounded below 1%. A settings-torus protocol reaches $5σ$ sensitivity at 0.1% within hours: a flat result forces $M\gtrsim 95\%$ of Hall's floor; a fringe would contradict the flat-rate prediction of quantum mechanics. Finally I exhibit a local mechanism -- moire phase locking -- with deterministic phase evolution and all randomness quenched in frozen lattice offsets and flight times, which attains the certified floor exactly at $η_{\rm eff}=1$ (10% uniform fidelity); its softening of the correlation extremes is a falsifiable fingerprint.

• The paper derives and certifies at key points the trade-off between measurement dependence, CHSH violation, and detector efficiency, including an exact quantum-point value of M=(27√2−33)/100 at S=2√2 and η=0.9.
• The paper shows that pure-detection local models reproducing singlet correlations must produce angle-dependent coincidence rates, with a predicted fourfold-sum fringe of 12.4% and a detection profile whose edge scales as √m.
• The paper presents moiré phase locking as an explicit saturating mechanism and proposes a full-setting-torus Fourier scan that can distinguish apparatus factorization from the predicted fringe at sub-percent sensitivity.

Overview and scope

This paper quantifies, with certified exactness at key points, the minimum measurement dependence required for a local hidden-variable (LHV) model to reproduce the phenomenology of a CHSH experiment on a polarization singlet. The setting is deliberately stricter than most relaxed-Bell analyses: models must be faithful, meaning they reproduce not only the coincidence correlators but also the observed singles rates, coincidence rates ($\eta^2$ per pair), and vanishing marginals of a maximally entangled state with symmetric detector efficiency $\eta$. The central object is the trade-off surface $M(S,\eta)$, where $M$ is Hall's variational measure of measurement dependence (2608.18886). The paper then follows the detection-sector mathematics to two consequences: an impossibility theorem forcing any pure-detection account into a measurable $\cos 4(a-b)$ fringe in the fourfold coincidence sum, and an explicit saturating mechanism ("moiré phase locking") whose residual correlation pattern constitutes a falsifiable fingerprint.

The price surface

For deterministic local strategies — each party assigns one element of $\{+1,-1,0\}$ per setting, giving $3^2 \times 3^2 = 81$ joint strategies and four distributions $q_{ab}$ over them — the minimization over the faithful polytope subject to the CHSH value equalling $S$ is a linear program in $4 \times 81$ variables. The paper conjectures the closed form

$\eta$0

The status of this claim is stated carefully: it is verified against the exact LP to $\eta$1 at 32 grid points spanning $\eta$2, $\eta$3; certified exactly at the quantum point and six rational points; but the general statement awaits a parametric dual certificate, so it remains formally a conjecture.

Three edges of the surface reproduce known theorems: at $\eta$4 it is Hall's tight line $\eta$5 [Hall 2011]; the zero set is the postselection ceiling $\eta$6 [Garg–Mermin 1987]; and at $\eta$7 the section vanishes exactly at the Garg–Mermin detection threshold $\eta$8. A notable corollary is that $\eta$9 whenever $M(S,\eta)$0: ordinary classical statistics implies no measurement dependence, so only correlations in the violating region force it. The inverted form,

$M(S,\eta)$1

shows that the measurement-dependence budget is amplified by $M(S,\eta)$2, i.e., acts on the coincidence subensemble. Importantly, the naive additive hypothesis sharing both endpoints is falsified — the true surface lies strictly below it in the interior.

Exact certification at the quantum point

Theorem (quantum point). At $M(S,\eta)$3, $M(S,\eta)$4, the minimal measurement dependence of faithful local models is exactly $M(S,\eta)$5.

The proof pipeline uses floating-point LP solves only to propose candidates: every nonzero primal weight (68 total) and dual multiplier is reconstructed in $M(S,\eta)$6 by integer-relation detection and verified symbolically, giving matching primal and dual certificates in exact arithmetic. At the optimum all six pairwise variational distances between the conditional distributions are equal — the extremal model distributes its measurement dependence symmetrically across setting contexts, a symmetry expected to organize the parametric dual family. The same pipeline yields exact rational values at six further $M(S,\eta)$7 points, each matching the conjectured formula. These certificates fix the price exactly rather than numerically, which distinguishes the result from grid-verified LP studies.

The exact detection sector

For deterministic sign outcomes with probabilistic detection (shared phase $M(S,\eta)$8, click probability $M(S,\eta)$9 with $M$0), the coincidence correlator becomes a ratio of circular autocorrelations, and exact reproduction of the singlet reduces to the functional equation $M$1 for all $M$2. Two analytic results anchor the numerical solution:

• Edge law: any continuous solution satisfies $M$3 as $M$4, obtained by matching power-law scalings of sign-disagreement arcs.
• Impossibility (pure detection): no profile $M$5 yields both $M$6 and an angle-independent coincidence rate. Angle-independent rates force $M$7 constant, which produces the sawtooth correlation instead of the cosine.

Solving the functional equation with the $M$8 edge law built in gives maximal residuals of $M$9, with the correlation curve matching the quantum prediction to $\cos 4(a-b)$0. The smooth factor $\cos 4(a-b)$1 in $\cos 4(a-b)$2 has $\cos 4(a-b)$3 but no identified closed form (the candidate $\cos 4(a-b)$4 is excluded at $\cos 4(a-b)$5) — an acknowledged open question. The profile's maximal-scale efficiency is $\cos 4(a-b)$6, consistent with sitting below the Garg–Mermin ceiling as required. A related structural argument via Schoenberg's theorem shows that whenever coincidence rates are flat, the correlator is positive definite and Tsirelson's bound holds for all angles; adversarial search over 300 random positive harmonic mixtures never exceeded $\cos 4(a-b)$7. Super-quantum postselected values are purchased with angle-dependent rates — a hard edge-band model reaching $\cos 4(a-b)$8 shows 48% rate modulation.

Moiré phase locking: an explicit saturating mechanism

The paper exhibits an explicit member of the surviving LHV space that attains the certified floor. In "moiré phase locking," a hidden phase evolves deterministically through the standing beat pattern between the pair's internal structure and the analyzers, with no stochastic dynamics; all randomness is quenched — frozen lattice offsets and flight times drawn once from a setting-independent density. Weights are selected by greedy search (23 active beat terms) and the ensemble by linear programming over offset grids.

At $\cos 4(a-b)$9, $\{+1,-1,0\}$0, with the correlation curve held uniformly within tolerance $\{+1,-1,0\}$1 of the quantum cosine:

Uniform tolerance $\{+1,-1,0\}$2	0.10	0.05	0.02	0.01
$\{+1,-1,0\}$3	1.000	1.015	1.235	1.375

At $\{+1,-1,0\}$4 the floor is attained exactly. The prices are cross-validated by independently coded solvers to the displayed precision, but the paper is explicit that they are properties of the stated discrete-time map, not continuum limits: halving the timestep shifts the $\{+1,-1,0\}$5 price from 1.375 toward $\{+1,-1,0\}$6 by first-order extrapolation. Two limitations are conceded plainly: frozen offsets break rotational covariance (the same solution evaluated on an unconstrained analyzer slice yields residuals up to 0.065, and a rotation-invariant register is left to future work), and the optimizer's freedom in choosing $\{+1,-1,0\}$7 is a non-dynamical ingredient whose derivation from a formation process remains an open problem. Composed with the solved detection profile via a one-parameter reweighting, the combined system interpolates from the source-informed channel to zero measurement dependence at $\{+1,-1,0\}$8, consistent with the detection-only channel's certified ceiling of $\{+1,-1,0\}$9. The residual pattern — softened correlation extremes concentrated between the Bell angles — is proposed as a falsifiable fingerprint at the $3^2 \times 3^2 = 81$0 level.

The fringe and its experimental status

For the exact profile, the total fourfold-sum coincidence rate carries a $3^2 \times 3^2 = 81$1-dominated fringe of relative peak-to-peak amplitude 12.4%, with leading harmonic exceeding its first overtone roughly 6:1. Hybrid models paying part of their bill through measurement dependence show proportionally reduced amplitude: along the interpolating detection family, bounding the fringe below 2%, 1%, and 0.5% forces $3^2 \times 3^2 = 81$2 to at least 45%, 68%, and 80% of Hall's floor respectively (family-dependent upper bounds on the detection share). Critically, the hiding theorem shows the fringe vanishes identically at the CHSH angles, term by term for every harmonic — standard four-point experiments are mathematically blind to it.

A survey of the observational record supports the paper's claim that this observable has never been bounded at sub-percent precision: the Aspect experiment reported the sum only as "typically" $3^2 \times 3^2 = 81$3; anomalies reported in raw Innsbruck data were contested and deflected rather than excluded; and the modern loophole-free tests adopted Eberhard/CH forms that rigorously sidestep fair sampling while leaving the observable unmeasured. The single-channel Brida et al. scan is distinguished as a near-miss that bounds nothing about the two-channel fourfold sum.

Protocol: factorization veto

The proposed protocol scans the full settings torus (e.g., $3^2 \times 3^2 = 81$4) and Fourier-analyzes $3^2 \times 3^2 = 81$5. Factorizable apparatus response $3^2 \times 3^2 = 81$6 is additive after the logarithm and therefore occupies only the Fourier axes, at any amplitude or drift phase; the physical $3^2 \times 3^2 = 81$7 signal occupies anti-diagonal pixels $3^2 \times 3^2 = 81$8 exclusively. Simulations with 3% apparatus ripple and Poisson noise recover null inputs as $3^2 \times 3^2 = 81$9, and Poisson-limited $q_{ab}$0 sensitivity at $q_{ab}$1 requires about 7 hours at $q_{ab}$2 pairs/s, or 20 minutes at $q_{ab}$3 pairs/s. Both kill conditions are informative: flatness at 0.1% converts through the surface into a mandatory floor of $q_{ab}$4 of Hall's bound, making measurement dependence the governing framework for any surviving local realism; a fringe of the predicted shape falsifies a flat-rate prediction of quantum mechanics itself; a fringe of the wrong shape excludes the detection sector as constructed. The paper notes that systematics beyond those simulated require standard experimental care.

Limitations and open questions

The paper states its own boundaries explicitly: the general price surface is machine-verified but lacks a parametric dual certificate; the guilt trade-off is computed along one specific detection family and provides upper bounds on the true detection share; the detection-profile factor $q_{ab}$5 lacks a closed form; the moiré prices depend on the discrete-time map rather than the underlying flow; and the rotation-invariant register and derivation of the quenched ensemble density $q_{ab}$6 remain open.

Conclusion

This paper converts the qualitative landscape of relaxed Bell assumptions into a priced object: an exact (at certified points) trade-off between measurement dependence and detector efficiency under full faithfulness, saturated by an explicit deterministic mechanism, and connected to a concrete, historically unmeasured observable — the angular modulation of the fourfold coincidence sum — that vanishes at precisely the settings used in every standard CHSH test. The proposed torus-scan protocol renders the resulting dichotomy experimentally decidable within hours on existing hardware.